Integrating a Spherical Coordinate Problem

[yaho8888]

Homework Statement

Evaluate the integral below, where H is the solid hemisphere x^2 + y^2 + z^2 ≤ 9, z ≤ 0

$$\iiint 8-x^2-y^2\,dx\,dy\,dz$$.

Homework Equations

none

The Attempt at a Solution

$$\int_{0}^{2\pi} \int_{\frac{\pi}{2}}^{\pi} \int_{0}^{3} (8-2p^2 \sin^2{\phi}) p^2 \sin{\phi}\ ,dp\,d\phi\, d\theta$$

do I set up the integral right?

 

[HallsofIvy]

In polar coordinates $x= \rho cos(\theta)sin(\phi)$ and $y= \rho sin(\theta)sin(\phi)$ so
$$x^2+ y^2= \rho^2 cos^2(\theta) sin^2(\phi)+ \rho^2 sin^2(\theta) sin^2(\phi)$$
$$= \rho^2 sin^2(\phi)(cos^2(\theta)+ sin^2(\theta))= \rho^2 sin^2(\phi)$$
NOT "$2\rho^2 sin^2(\phi)$.

 

[yaho8888]

ye! sorry. 8-p^2*sin^2(phi)

but if I use $$= \rho^2 sin^2(\phi)$$ are the limit of this integral right for the question stated above.